On a problem of Erd\H{o}s and Rothschild on edges in triangles

Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which is contained in at least one triangle, must contain an edge that is in at least H(N,C) triangles. In particular, Erd\H{o}s asked in 1987 to determine whether for every C>0 there is \epsilon >0 such that H(N,C) > N^\epsilon, for all sufficiently large N. We prove that H(N,C) = N^{O(1/log log N)} for every fixed C < 1/4. This gives a negative answer to the question of Erd\H{o}s, and is best possible in terms of the range for C, as it is known that every N-vertex graph with more than (N^2)/4 edges contains an edge that is in at least N/6 triangles.Comment: 8 page

High-resolution absorption spectroscopy of the circumgalactic medium of the Milky Way

In this article we discuss the importance of high-resolution absorption spectroscopy for our understanding of the distribution and physical nature of the gaseous circumgalactic medium (CGM) that surrounds the Milky Way. Observational and theoretical studies indicate a high complexity of the gas kinematics and an extreme multi-phase nature of the CGM in low-redshift galaxies. High-precision absorption-line measurements of the Milky Way's gas environment thus are essential to explore fundamental parameters of circumgalactic gas in the local Universe, such as mass, chemical composition, and spatial distribution. We shortly review important characteristics of the Milky Way's CGM and discuss recent results from our multi-wavelength observations of the Magellanic Stream. Finally, we discuss the potential of studying the warm-hot phase of the Milky Way's CGM by searching for extremely weak [FeX] l6374.5 and [FeIVX] l5302.9 absorption in optical QSO spectra.Comment: 7 pages, 4 figures; accepted for publication in Astronomical Notes (paper version of a talk presented at the 10th Thinkshop, Potsdam, 2013

Numerical Description of Dilute Particle-Laden FLows by a Quadrature-Based Moment Method

The numerical simulation of gas-particle flows is divided into two families of methods. In Euler-Lagrange methods individual particle trajectories are computed, whereas in Euler-Euler methods particles are characterized by statistical descriptors. Lagrangian methods are very precise but their computational cost increases with instationarity and particle volume fraction. In Eulerian methods (also called moment methods) the particle-phase computational cost is comparable to that of the fluid phase but requires strong simplificaions. Existing Eulerian models consider unimodal or close-to-equilibrium particle velocity distributions and then fail when the actual distribution is far from equilibrium. Quadrature-based Eulerian methods introduce a new reconstruction of the velocity distribution, written as a sum of delta functions in phase space constrained to give the right values for selected low-order moments. Two of the quadrature-based Eulerian methods, differing by the reconstruction algorithm, are the focus of this work. Computational results for two academic cases (crossing jets, Taylor-Green flow) are compared to those of a Lagrangian method (considered as the reference solution) and of an existing second-order moment method. With the quadrature-based Eulerian methods, significant qualitative improvement is noticed compared to the second-order moment method in the two test cases

A Survey of O VI, C III, and H I in Highly Ionized High-Velocity Clouds

(ABRIDGED) We present a Far-Ultraviolet Spectroscopic Explorer survey of highly ionized high-velocity clouds (HVCs) in 66 extragalactic sight lines. We find a total of 63 high-velocity O VI absorbers, 16 with 21 cm-emitting H I counterparts and 47 ``highly ionized'' absorbers without 21 cm emission. 11 of these high-velocity O VI absorbers are positive-velocity wings (broad O VI features extending asymmetrically to velocities of up to 300 km/s). The highly ionized HVC population is characterized by =38+/-10 km/s and <log N_a(O VI)>=13.83+/-0.36. We find that 81% (30/37) of high-velocity O VI absorbers have clear accompanying C III absorption, and 76% (29/38) have accompanying H I absorption in the Lyman series. The lower average width of the high-velocity H I absorbers implies the H I lines arise in a separate, lower temperature phase than the O VI. We find that the shape of the wing profiles is well reproduced by a radiatively cooling, vertical outflow. However, the outflow has to be patchy and out of ionization equilibrium. An alternative model, consistent with the observations, is one where the highly ionized HVCs represent the low N(H I) tail of the HVC population, with the O VI formed at the interfaces around the embedded H I cores. Though we cannot rule out a Local Group explanation, we favor a Galactic origin. This is based on the recent evidence that both H I HVCs and the million-degree gas detected in X-ray absorption are Galactic phenomena. Since the highly ionized HVCs appear to trace the interface between these two Galactic phases, it follows that highly ionized HVCs are Galactic themselves. However, the non-detection of high-velocity O VI in halo star spectra implies that any Galactic high-velocity O VI exists at z-distances beyond a few kpc.Comment: 36 pages, 14 figures (3 in color), accepted to ApJS. Some figures downgraded to limit file siz

On two problems in graph Ramsey theory

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a graph H is the least positive integer N such that every two-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of H. A famous result of Chv\'atal, R\"{o}dl, Szemer\'edi and Trotter states that there exists a constant c(\Delta) such that r(H) \leq c(\Delta) n for every graph H with n vertices and maximum degree \Delta. The important open question is to determine the constant c(\Delta). The best results, both due to Graham, R\"{o}dl and Ruci\'nski, state that there are constants c and c' such that 2^{c' \Delta} \leq c(\Delta) \leq 2^{c \Delta \log^2 \Delta}. We improve this upper bound, showing that there is a constant c for which c(\Delta) \leq 2^{c \Delta \log \Delta}. The induced Ramsey number r_{ind}(H) of a graph H is the least positive integer N for which there exists a graph G on N vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of H. Erd\H{o}s conjectured the existence of a constant c such that, for any graph H on n vertices, r_{ind}(H) \leq 2^{c n}. We move a step closer to proving this conjecture, showing that r_{ind} (H) \leq 2^{c n \log n}. This improves upon an earlier result of Kohayakawa, Pr\"{o}mel and R\"{o}dl by a factor of \log n in the exponent.Comment: 18 page

Multiphase Plasma in Sub-Damped Lyman Alpha Systems: A Hidden Metal Reservoir

We present a VLT/UVES spectrum of a proximate sub-damped Lyman-alpha (sub-DLA) system at z=2.65618 toward the quasar Q0331-4505 (z_qso=2.6785+/-0.0030). Absorption lines of O I, Si II, Si III, Si IV, C II, C III, C IV, Fe II, Al II, and O VI are seen in the sub-DLA, which has a neutral hydrogen column density log N(H I)=19.82+/-0.05. The absorber is at a velocity of 1820+/-250 km/s from the quasar; however, its low metallicity [O/H]=-1.64+/-0.07, lack of partial coverage, lack of temporal variations between observations taken in 2003 and 2006, and non-detection of N V imply the absorber is not a genuine intrinsic system. By measuring the O VI column density and assuming equal metallicities in the neutral and ionized gas, we determine the column density of hot ionized hydrogen in this sub-DLA, and in two other sub-DLAs with O VI drawn from the literature. Coupling this with determinations of the typical amount of warm ionized hydrogen in sub-DLAs, we confirm that sub-DLAs are a more important metal reservoir than DLAs, in total comprising at least 6-22% of the metal budget at z~2.5.Comment: 5 pages, 3 color figures, accepted for publication in ApJ

Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \epsilon may require as many as 2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page

An HST/COS legacy survey of high-velocity ultraviolet absorption in the Milky Way's circumgalactic medium and the Local Group

To characterize the absorption properties of this circumgalactic medium (CGM) and its relation to the LG we present the so-far largest survey of metal absorption in Galactic high-velocity clouds (HVCs) using archival ultraviolet (UV) spectra of extragalactic background sources. The UV data are obtained with the Cosmic Origins Spectrograph (COS) onboard the Hubble Space Telescope (HST) and are supplemented by 21 cm radio observations of neutral hydrogen. Along 270 sightlines we measure metal absorption in the lines of SiII, SiIII, CII, and CIV and associated HI 21 cm emission in HVCs in the velocity range |v_LSR|=100-500 km s^-1. With this unprecedented large HVC sample we were able to improve the statistics on HVC covering fractions, ionization conditions, small-scale structure, CGM mass, and inflow rate. For the first time, we determine robustly the angular two point correlation function of the high-velocity absorbers, systematically analyze antipodal sightlines on the celestial sphere, and compare the absorption characteristics with that of Damped Lyman alpha absorbers (DLAs) and constrained cosmological simulations of the LG. Our study demonstrates that the Milky Way CGM contains sufficient gaseous material to maintain the Galactic star-formation rate at its current level. We show that the CGM is composed of discrete gaseous structures that exhibit a large-scale kinematics together with small-scale variations in physical conditions. The Magellanic Stream clearly dominates both the cross section and mass flow of high-velocity gas in the Milky Way's CGM. The possible presence of high-velocity LG gas underlines the important role of the local cosmological environment in the large-scale gas-circulation processes in and around the Milky Way (abridged).Comment: 37 pages, 25 figures, 8 tables, accepted for publication in A&